## DETERMINANTS OF CENTRALIZABLE MATRICES OVER ANY SKEW-FIELD

**Xie Bangjie (Department of Mathematics)**

A n-rowed square matrix A over a skew-field K with central field F is said to be centralizable if the characteristic matrix λI—A can be reduced by some elementary transformations into the following diagonal form: such that φ_1(λ)|φ_2(λ)|…|φ_s(λ), where these φ_i(λ) (i=1,2,…,s) are all monic polynomials over F. For centralizable matrix A of form (1), the determinant of A cart be defined as follows: ‖A‖=(-1)~nφ_1(0)φ_2(0)…φ_3(0).In this paper some basic properties about determinants of centralizable matrices over K are considered.Some important matrices of quaternions are centralizable, for example, self-conjugate matrices of quaternions, generalized Ⅱ-matrices etc.Some theorems about real symmetrical matrices and Hermitian matrices are generalized also.

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